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Large Deviations and Gallavotti–Cohen Principle for Dissipative PDEs with Rough Noise

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Abstract

We study a class of dissipative PDEs perturbed by an unbounded kick force. Under some natural assumptions, the restrictions of solutions to integer times form a homogeneous Markov process. Assuming that the noise is rough with respect to the space variables and has a non-degenerate law, we prove that the system in question satisfies a large deviation principle (LDP) in τ-topology. Under some additional hypotheses, we establish a Gallavotti–Cohen type symmetry for the rate function of an entropy production functional and the strict positivity and finiteness of the mean entropy production rate in the stationary regime. The latter result is applicable to PDEs with strong nonlinear dissipation.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada

    V. Jakšić

  2. Laboratoire de Matématiques, UMR CNRS 8100, Université de Versailles-Saint-Quentin-en-Yvelines, 78035, Versailles, France

    V. Nersesyan

  3. Aix Marseille Université, CNRS, CPT, UMR 7332, Case 907, 13288, Marseille, France

    C.-A. Pillet

  4. Univeristé de Toulon, CNRS, CPT, UMR 7332, 83957, La Garde, France

    C.-A. Pillet

  5. Department of Mathematics, University of Cergy-Pontoise, CNRS UMR 8088, 2 Avenue Adolphe Chauvin, 95302, Cergy-Pontoise, France

    A. Shirikyan

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  1. V. Jakšić
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  2. V. Nersesyan
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  3. C.-A. Pillet
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  4. A. Shirikyan
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Correspondence to V. Jakšić.

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Communicated by H. Spohn

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Jakšić, V., Nersesyan, V., Pillet, CA. et al. Large Deviations and Gallavotti–Cohen Principle for Dissipative PDEs with Rough Noise. Commun. Math. Phys. 336, 131–170 (2015). https://doi.org/10.1007/s00220-014-2279-3

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  • DOI: https://doi.org/10.1007/s00220-014-2279-3

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